Important Points: Work and Energy

Table of Contents
1. Introduction
2. Concept of Work
3. Work Done by a Constant Force
4. Energy
5. Kinetic Energy
6. Potential Energy
7. Law of Conservation of Energy
8. Power - Rate of Doing Work
9. Examples and Applications
10. Key Learnings
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Introduction

• The chapter focuses on the concepts of work, energy and power, which are key to understanding many natural phenomena and everyday activities.
• Energy is crucial for survival and for performing various activities; it is obtained from food and other sources and can be transformed from one form to another.

Concept of Work

• Work in physics is different from everyday usage; it requires both a force acting on an object and a displacement of the object due to that force.
• Scientifically, work is done when a force causes displacement of the point of application of the force.
• Examples: pushing a pebble along the ground, pulling a trolley so it moves, and lifting a book from a table to a shelf all involve work as per the scientific definition.
• No work is done if either the force is absent or there is no displacement of the object in the direction of the force.
• Work is a scalar quantity - it has magnitude but no direction. Its SI unit is the joule (J).

Work Done by a Constant Force

• If a constant force F acts on an object and the object is displaced by s, and θ is the angle between the force and the displacement, then the work done is W = F s cosθ.
• When force and displacement are in the same direction, cosθ = 1 and W = F s (positive work).
• When force acts opposite to displacement, cosθ = -1 and the work is negative (force removes energy from the object).
• If the force is perpendicular to the displacement, cosθ = 0 and the work done by that force is zero (for example, when centripetal force acts perpendicular to instantaneous displacement in uniform circular motion).

Energy

• Energy is the capacity to do work. Objects that can do work are said to possess energy.
• Energy can be transferred between objects and transformed from one form to another (for example, chemical energy in food → mechanical energy in muscles).
• Common forms of energy: mechanical (kinetic and potential), heat, chemical, electrical, and light energy.
• The SI unit of energy is the joule (J), the same as for work.

Kinetic Energy

• Kinetic energy is the energy possessed by a body due to its motion.
• The kinetic energy of a body of mass m moving with speed v is given by KE = 1/2 m v².

Derivation of the kinetic energy formula (for constant force)

The following derivation shows how work done by a constant force changes the kinetic energy of a body.

Let a constant force act on a body of mass m and produce an acceleration a, causing its speed to change from u to v while it is displaced by s.

Work done by the force is W = F s.

By Newton's second law, F = m a.

Therefore, W = m a s.

Using the kinematic relation v² - u² = 2 a s, we can write a s = (v² - u²)/2.

Substituting gives W = m × (v² - u²)/2.

Thus, the work done on the body equals the change in its kinetic energy: W = 1/2 m v² - 1/2 m u².

If the body starts from rest (u = 0), the kinetic energy acquired is KE = 1/2 m v².

Potential Energy

• Potential energy is stored energy due to an object's position or configuration.
• Examples: a stretched rubber band, a wound-up toy, or a book raised to a shelf all have stored energy.
• The gravitational potential energy of a body of mass m at height h above a chosen reference level (usually the ground) is PE = m g h, where g is the acceleration due to gravity.
• Elastic potential energy stored in a spring obeying Hooke's law is PE_spring = 1/2 k x², where k is the spring constant and x is the extension or compression from the equilibrium position.

Law of Conservation of Energy

• The law of conservation of energy states that energy cannot be created or destroyed; it can only be transformed from one form to another.
• The total energy of an isolated system remains constant before and after any transformation.
• Example: In free fall, an object's gravitational potential energy is converted into kinetic energy; the sum of potential and kinetic energies (mechanical energy) remains the same if no non-conservative forces (like friction or air resistance) do work.
• In real situations, non-conservative forces such as friction convert mechanical energy into other forms (typically thermal energy), so mechanical energy is not conserved but the total energy still is.

Power - Rate of Doing Work

• Power is the rate at which work is done or energy is transferred.
• The SI unit of power is the watt (W), where 1 W = 1 J s⁻¹.
• If W joules of work are done in t seconds, average power is P = W / t.
• For a constant force moving an object with constant velocity v, power can also be written as P = F v, because work per unit time equals force times velocity in the direction of force.
• Power depends on how fast work is done: the same amount of work done in less time requires greater power.

Examples and Applications

• Lifting a book: Work done = m g h where m is mass of the book and h is the height raised; the energy spent converts chemical energy in muscles into gravitational potential energy.
• Pushing a trolley horizontally: If the applied force is parallel to displacement, work done = F s.
• Carrying an object horizontally at constant height: The upward force applied by muscles does no work with respect to the horizontal displacement because it is perpendicular to displacement; however, the person still expends metabolic energy due to internal muscular activity and friction in the body.
• Electrical appliances: The power rating (in watts) on appliances indicates how much energy they use per second; a 60 W bulb uses 60 joules of electrical energy each second it is on.

Key Learnings

• The scientific definition of work involves both force and displacement.
• Energy is the capacity to do work and exists in various forms, notably kinetic and potential for mechanical systems.
• The law of conservation of energy states that total energy remains constant in an isolated system while energy may be transformed from one form to another.
• Power quantifies how quickly work is done or energy is transformed.